Figure 2 - uploaded by Manuel Winkler
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a) Polar angle correspondence for points ζ m in the ζ-plane based on the normalized path coordinate s of points zm in the the z-plane and b) Update of tunnel boundary points zm based on the principle of corresponding point polar angle Equality (PCPPAE) after He et al. (2022) by polar projection of points zc,m onto boundary C within a single iteration step j (Exemplarily shown for three points with varying index m). í µí±í µí±(í µí± í µí± , í µí±í µí± í µí±í µí± ) = � � �í µí±¥í µí±¥ í µí±í µí± − í µí±¥í µí±¥ í µí±í µí±,í µí±í µí± (í µí± í µí± , í µí±í µí± í µí±í µí± )� 2 + �í µí±¦í µí±¦ í µí±í µí± − í µí±¦í µí±¦ í µí±í µí±,í µí±í µí± (í µí± í µí± , í µí±í µí± í µí±í µí± )� 2 í µí±í µí± í µí±í µí±=1
Source publication
In case of tunnels with arbitrary geometries, solutions for stresses and displacements in the tunnel exterior might be derived with the aid of the conformal mapping technique of the complex variable method. Thereby, the physical tunnel domain is mapped onto a fictitious unit circle domain on which the elastic potentials, as part of the final soluti...
Contexts in source publication
Context 1
... order to solve for the mapping coefficients, the tunnel contour C in the z-plane needs to be discretized into a finite number of M points zm (m = 1, 2,…, M). For an exemplarily chosen semicircular cross-section, an increased point density in corners of boundary C on the z-plane is used as suggested by Exadaktylos et al. (2003) for regions with large variations of the radius of curvature (Figure 2a). Each of the discrete points zm from the z-plane is associated with a corresponding point ζm (m = 1, 2,…, M) on the tunnel contour in the ζ-plane for which the position is not known in advance. ...
Context 2
... polar angles θm are equated with ϑm= sm/2π+ϑoff, describing the normalized path coordinate sm of a corresponding point zm along the initially discretized tunnel boundary C including a considered offset angle ϑoff equal to the polar angle of zm=1. As can be seen in Figure 2a), the path coordinate s is chosen to start from z1 at the symmetry axis of the considered semi-circular cross section. ...
Context 3
... in each iteration step j, the points zm j are updated by projecting the image points zc,m j-1 onto the physical tunnel boundary C via projection lines connecting points zc,m j-1 with the coordinate origin. A graphical representation of this principle is given in Figure 2b). The points obtained are used to replace the left-hand side in Eqn. ...
Citations
... Consequently, in this paper a solution strategy to overcome such difficulties in connection with complex variable solutions is provided. The assumption is made that the mapping coefficients of the conformal mapping function are already known, e.g. from an iterative scheme as presented by Winkler et al. (2023). Finally, the solutions for stresses and displacements of a specific case are compared with the results from finite element calculations. ...
The implementation of closed-form solutions for stress and displacement fields around tunnels with arbitrary geometry, often based on the complex variable theory and the method of conformal mapping, can be quite challenging from a mathematical point of view. In this paper a solution strategy for the implementation of a chosen closed-form solution from literature is presented, including the possibility to account for rock mass anisotropy and arbitrary tunnel geometries. The evaluation of the involved elastic potential functions is described, respectively derivatives thereof, in terms of solving non-linear constrained optimization problems. To validate our approach, the analytical results for stresses and displacements around a tunnel with semicircular geometry are compared to numerical results from finite element computations. The outcome of the study should be regarded as a basis for the development of refined analytical solutions within anisotropic rock masses considering more realistic boundary conditions and effects such as material non-linearity.
